Q.Repot very damaged old Christmas cactus.
I’ve included pictures of what my Grandma’s Christmas cactus has been through this past year. A move from NC to TX, very dry out and wind damaged, and then over watered. Some of these things were out of my control. I fortunately have been able to get some cuttings started so I haven’t totally lost my beloved plant. Please help me to save the rest of it.

Certified GKH Gardening Expert
If it was overwatered, be sure to repot in cacti or succulent mix. As you suggested, try to propagate from healthy tips. That is probably your best bet. The original doesn't look too good. You can try cutting it back as far as you can, still leaving a few leaves. It may bounce back. They are pretty tough.